This document provides an overview of exponent rules, including: - When multiplying or dividing terms with the same base, you add or subtract their exponents respectively - When raising a power to another power, you multiply the exponents - A product or quotient raised to a power means each term is raised to that power - A number raised to the zero power is equal to 1, except for 0 raised to zero which is undefined - A negative exponent flips the term with that exponent to the denominator Several examples are given to demonstrate each rule and mixed practice problems are provided for application.

1. 11

2. Exponent RulesExponent Rules

3. PartsParts
 When a number, variable, or expression isWhen a number, variable, or expression is
raised to a power, the number, variable, orraised to a power, the number, variable, or
expression is called theexpression is called the basebase and the power isand the power is
called thecalled the exponentexponent..
n
b

4. What is an Exponent?What is an Exponent?
 An exponent means that you multiply the baseAn exponent means that you multiply the base
by itself that many times.by itself that many times.
 For exampleFor example
xx44
== xx ●● xx ●● xx ●● xx
2266
== 22 ●● 22 ●● 22 ●● 22 ●● 22 ●● 22 = 64= 64

5. The Invisible ExponentThe Invisible Exponent
 When an expression does not have a visibleWhen an expression does not have a visible
exponent its exponent is understood to be 1.exponent its exponent is understood to be 1.
1
xx =

6. Exponent Rule #1Exponent Rule #1
 WhenWhen multiplyingmultiplying two expressions with thetwo expressions with the
same base yousame base you addadd their exponents.their exponents.
 For exampleFor example
=⋅ mn
bb mn
b +
=⋅ 42
xx =+42
x 6
x
=⋅ 2
22 21
22 ⋅ 21
2 +
= 3
2= 8=

7. Exponent Rule #1Exponent Rule #1
 Try it on your own:Try it on your own:
=⋅ mn
bb mn
b +
=⋅ 73
.1 hh
=⋅33.2 2
1073
hh =+
312
33 =+
27333 =⋅⋅=

8. Exponent Rule #2Exponent Rule #2
 WhenWhen dividingdividing two expressions with the sametwo expressions with the same
base youbase you subtractsubtract their exponents.their exponents.
 For exampleFor example
=m
n
b
b mn
b −
=2
4
x
x
=−24
x 2
x

9. Exponent Rule #2Exponent Rule #2
 Try it on your own:Try it on your own:
=m
n
b
b mn
b −
=2
6
.3
h
h
=
3
3
.4
3
=−26
h 4
h
=−13
3 =2
3 9

10. Exponent Rule #3Exponent Rule #3
 When raising aWhen raising a power to a powerpower to a power youyou
multiplymultiply the exponentsthe exponents
 For exampleFor example
mn
b )( mn
b ⋅
=
42
)(x 42⋅
= x 8
x=
22
)2( 22
2 ⋅
= 4
2= 16=

11. Exponent Rule #3Exponent Rule #3
 Try it on your ownTry it on your own
mn
b )( mn
b ⋅
=
23
)(.5 h 23⋅
= h 6
h=
22
)3(.6 22
3 ⋅
= 4
3= 81=

12. NoteNote
 When using this rule the exponent can not beWhen using this rule the exponent can not be
brought in the parenthesisbrought in the parenthesis if there is additionif there is addition
or subtractionor subtraction
222
)2( +x 44
2+x≠
You would have to use FOIL in these cases

13. Exponent Rule #4Exponent Rule #4
 When a product is raised to a power, eachWhen a product is raised to a power, each
piece is raised to the powerpiece is raised to the power
 For exampleFor example
m
ab)( mm
ba=
2
)(xy 22
yx=
2
)52( ⋅ 22
52 ⋅= 254⋅= 100=

14. Exponent Rule #4Exponent Rule #4
 Try it on your ownTry it on your own
m
ab)( mm
ba=
3
)(.7 hk 33
kh=
2
)32(.8 ⋅ 22
32 ⋅= 94⋅= 36=

15. NoteNote
 This rule is for products only. When using thisThis rule is for products only. When using this
rule the exponent can not be brought in therule the exponent can not be brought in the
parenthesisparenthesis if there is addition orif there is addition or
subtractionsubtraction
2
)2( +x 22
2+x≠
You would have to use FOIL in these cases

16. Exponent Rule #5Exponent Rule #5
 When a quotient is raised to a power, both theWhen a quotient is raised to a power, both the
numerator and denominator are raised to thenumerator and denominator are raised to the
powerpower
 For exampleFor example
=





m
b
a
m
m
b
a
=





3
y
x
3
3
y
x

17. Exponent Rule #5Exponent Rule #5
 Try it on your ownTry it on your own
=





m
b
a
m
m
b
a
=





2
.9
k
h
2
2
k
h
=





2
2
4
.10 2
2
2
4
4
16
= 4=

18. Zero ExponentZero Exponent
 When anything, except 0, is raised to the zeroWhen anything, except 0, is raised to the zero
power it is 1.power it is 1.
 For exampleFor example
=0
a 1 ( if a ≠ 0)
=0
x 1 ( if x ≠ 0)
=0
25 1

19. Zero ExponentZero Exponent
 Try it on your ownTry it on your own
=0
a 1 ( if a ≠ 0)
=0
.11 h 1 ( if h ≠ 0)
=0
1000.12 1
=0
0.13 0

20. Negative ExponentsNegative Exponents
 If bIf b ≠ 0, then≠ 0, then
 For exampleFor example
=−n
b n
b
1
=−2
x 2
1
x
=−2
3 2
3
1
9
1
=

21. Negative ExponentsNegative Exponents
 If bIf b ≠ 0, then≠ 0, then
 Try it on your own:Try it on your own:
=−n
b n
b
1
=−3
.14 h 3
1
h
=−3
2.15 3
2
1
8
1
=

22. Negative ExponentsNegative Exponents
 The negative exponent basically flips the partThe negative exponent basically flips the part
with the negative exponent to the other half ofwith the negative exponent to the other half of
the fraction.the fraction.






−2
1
b 





=
1
2
b 2
b=






−2
2
x 





=
1
2 2
x 2
2x=

23. Math MannersMath Manners
 For a problem to be completelyFor a problem to be completely
simplified there should not be anysimplified there should not be any
negative exponentsnegative exponents

24. Mixed PracticeMixed Practice
9
5
3
6
.1
d
d 95
2 −
= d 4
2 −
= d 4
2
d
=
54
42.2 ee 54
8 +
= e 9
8e=

25. Mixed PracticeMixed Practice
( )54
.3 q 54⋅
= q 20
q=
( )5
2.4 lp 555
2 pl= 55
32 pl=

26. Mixed PracticeMixed Practice
2
42
)(
)(
.5
xy
yx
22
48
yx
yx
= 2428 −−
= yx 26
yx=
9
253
)(
.6
x
xx
9
28
)(
x
x
= 9
16
x
x
= 916−
= x 7
x=

27. Mixed PracticeMixed Practice
6523246
)()(.7 pnmnm
301218812
pnmnm ⋅=
301281812
pnm ++
=
302030
pnm=

28. Mixed PracticeMixed Practice
4
6
)2(
)2(
.8
yx
yx
−
− 46
)2( −
−= yx 2
)2( yx −=
)2)(2( yxyx −−=
F
2
x=
O
xy2−
I
xy2−
L
2
4y+
22
44 yxyx +−=

29. Mixed PracticeMixed Practice
94
56
.9
da
da 9546 −−
= da 42 −
= da
4
2
d
a
=

30. 3030
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